4346 Each series satisfies the hypotheses of the alternating seriestest. Approximate the

What characterizes an alternating series?

(a) The series \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{2}}\) converges by the alternating series test since ________ and ________. (b) If \(S=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k^{2}}\) and \(s_{9}=\sum_{k=1}^{9}\frac{(-1)^{k+1}}{k^{2}}\) then \(\left|S-s_{9}\right|<\)_________. Equation Transcription: _____ Text Transcription: Sum over k=1 ^infty (-1)^k+1/k^2 S= sum over k=1 ^infty (-1)^k+1/k^2 s_9=sum over k=1 ^9 (-1)^k+1/k^2 |S-s_9|<_____

Classify each sequence as conditionally convergent, absolutely convergent, or divergent. (a) \(\sum_{k=1}^{\infty}(-1)^{k+1} \frac{1}{k}\) : ________. (b) \(\sum_{k=1}^{\infty}(-1)^{k} \frac{3 k-1}{9 k+15}\) : ________. (c) \(\sum_{k=1}^{s}(-1)^{k} \frac{1}{k(k+2)}\) : ________. (d) \(\sum_{k=1}^{\infty}(-1)^{k+1} \frac{1}{\sqrt[4]{k}^{3}}\) : ________. Equation Transcription: :_____ :_____ :_____ :_____ Text Transcription: Sum over k=1 ^infty (-1)^k+1 1/k:_____ Sum over k=1 ^infty (-1)^k 3k-1/9k+15:_____ Sum over k=1 ^infty (-1)^k 1/k(k+2):_____ Sum over k=1 ^infty (-1)^k+1 1/sqrt ^4_k^3:_____

Given that \(\lim _{k \rightarrow+\infty} \frac{(k+1)^{4} / 4^{k+1}}{k^{4} / 4^{k}}=\lim _{k \rightarrow+\infty} \frac{\left(1+\frac{1}{k}\right)^{4}}{4}=\frac{1}{4}\) is the series \(\sum_{k=1}^{\infty}(-1)^{k} k^{4} / 4^{k}\) conditionally convergent, absolutely convergent, or divergent? Equation Transcription: Text Transcription: lim over k rightarrow +infty (k+1)^4/4^k+1/k^4/4^k=lim over k rightarrow +infty (1 1/k)^4 4=1/4 Sum over k=1 ^infty (-1)^k k^4/4^k

Determine whether the alternating series converges; justify your answer. \(\sum_{k=1}^{=}(-1)^{k+1} e^{-k}\) Equation Transcription: Text Transcription: Sum over k=1 ^infty (-1)^k+1 e^-k

Determine whether the alternating series converges; justify your answer. \(\sum_{k=3}^{\infty}(-1)^{k} \frac{\ln k}{k}\) Equation Transcription: Text Transcription: Sum over k=1 ^infty (-1)^k ln k/k

Use the ratio test for absolute convergence (Theorem 9.6.5) to determine whether the series converges or diverges. If the test is inconclusive, say so. \(\sum_{k=1}^{\infty}\left(-\frac{3}{5}\right)^{k}\) Equation Transcription: Text Transcription: Sum over k=1 ^infty (-3/5)^k

Use the ratio test for absolute convergence (Theorem 9.6.5) to determine whether the series converges or diverges. If the test is inconclusive, say so. \(\sum_{k=1}^{\infty}(-1)^{k+1} \frac{2^{k}}{k !}\) Equation Transcription: Text Transcription: Sum over k=1 ^infty (-1)^k+1 2^k/k!

Use the ratio test for absolute convergence (Theorem 9.6.5) to determine whether the series converges or diverges. If the test is inconclusive, say so. \(\sum_{k=1}^{\infty}(-1)^{k+1} \frac{3^{k}}{k^{2}}\) Equation Transcription: Text Transcription: Sum over k=1 ^infty (-1)^k+1 3^k/k^2

Use the ratio test for absolute convergence (Theorem 9.6.5) to determine whether the series converges or diverges. If the test is inconclusive, say so. \(\sum_{k=1}^{\infty}(-1)^{k} \frac{k}{5^{k}}\) Equation Transcription: Text Transcription: Sum over k=1 ^infty (-1)^k k/5^k

Use the ratio test for absolute convergence (Theorem 9.6.5) to determine whether the series converges or diverges. If the test is inconclusive, say so. \(\sum_{k=1}^{\infty}(-1)^{k} \frac{k^{3}}{e^{k}}\) Equation Transcription: Text Transcription: Sum over k=1 ^infty (-1)^k k^3/e^k

Use the ratio test for absolute convergence (Theorem 9.6.5) to determine whether the series converges or diverges. If the test is inconclusive, say so. \(\sum_{k=1}^{\infty}(-1)^{k+1} \frac{k^{k}}{k !}\) Equation Transcription: Text Transcription: Sum over k=1 ^infty (-1)^k+1 k^k/k!

Classify each series as absolutely convergent, conditionally convergent, or divergent. \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{3 k}\) Equation Transcription: Text Transcription: Sum over k=1 ^infty (-1)^k+1/3k

Classify each series as absolutely convergent, conditionally convergent, or divergent. \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{4 / 3}} \) Equation Transcription: Text Transcription: sum_{k=1}^{infty} frac{(-1)^k+1}{k^{4/3}

Classify each series as absolutely convergent, conditionally convergent, or divergent. \(\sum_{k=1}^{\infty} \frac{(-4)^{k}}{k^{2}}\) Equation Transcription: Text Transcription: sum_{k=1}^{infty} frac{(-4)^k}{k^2}

Classify each series as absolutely convergent, conditionally convergent, or divergent. \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k !}\) Equation Transcription: Text Transcription: sum_{k=1}^{infty} frac{(-1)^k+1}{k!}

Classify each series as absolutely convergent, conditionally convergent, or divergent. \(\sum_{k=1}^{\infty} \frac{\cos k \pi}{k}\) Equation Transcription: Text Transcription: sum_{k=1}^{infty} frac{\cos k pi}{k}

Classify each series as absolutely convergent, conditionally convergent, or divergent. \(\sum_{k=3}^{\infty}\frac{(-1)^k\ln\ k}{k}\) Equation Transcription: Text Transcription: sum_{k=3}^{infty} frac{(-1)^{k} ln k}{k}

Classify each series as absolutely convergent, conditionally convergent, or divergent. \(\sum_{k=1}^{\infty}(-1)^{k+1}\ \frac{k+2}{k(k+3)}\) Equation Transcription: Text Transcription: sum_{k=1}^{infty}(-1)^k + 1} frac{k + 2}{k(k + 3}

Classify each series as absolutely convergent, conditionally convergent, or divergent. \(\sum_{k=1}^{\infty}\frac{(-1)^{k+1}\ k^2}{k^3+1}\) Equation Transcription: Text Transcription: sum_{k=1}^{infty} frac{(-1)^{k+1} k^2}{k^3+1}

Classify each series as absolutely convergent, conditionally convergent, or divergent. \(\sum_{k=1}^{\infty} \sin \frac{k \pi}{2}\) Equation Transcription: Text Transcription: sum_{k=1}^{infty} sin frac{kpi}{2}

Classify each series as absolutely convergent, conditionally convergent, or divergent. \(\sum_{k=1}^{\infty} \frac{\sin k}{k^{3}}\) Equation Transcription: Text Transcription: sum_{k=1}^{infty} frac{sin k}{k^3}

Classify each series as absolutely convergent, conditionally convergent, or divergent. \(\sum_{k=2}^{\infty}\frac{(-1)^k}{k\ln\ k}\) Equation Transcription: Text Transcription: sum_{k=2}^{infty} frac{(-1)^k}{k ln k}

Classify each series as absolutely convergent, conditionally convergent, or divergent. \(\sum_{k=1}^{\infty} \frac{(-1)^{k}}{\sqrt{k(k+1)}}\) Equation Transcription: Text Transcription: sum_{k=1}^{infty} frac{(-1)^k}{sqrt{k(k+1)}

Classify each series as absolutely convergent, conditionally convergent, or divergent. \(\sum_{k=2}^{\infty}\left(-\frac{1}{\ln\ k}\right)^{k}\) Equation Transcription: Text Transcription: sum_{k=2}^{infty} (-frac{1}{ln k})^k

Classify each series as absolutely convergent, conditionally convergent, or divergent. \(\sum_{k=1}^{\infty}\frac{k\ \cos\ k\pi}{k^2+1}\) Equation Transcription: Text Transcription: sum_{k=1}^{\infty} frac{k cos k pi}{k^2+1}

Classify each series as absolutely convergent, conditionally convergent, or divergent. \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1} k !}{(2 k-1) !}\) Equation Transcription: Text Transcription: sum_{k=1}^{infty} frac{(-1)^k+1 k!}{(2k - 1)!

Classify each series as absolutely convergent, conditionally convergent, or divergent. \(\sum_{k=1}^{\infty}(-1)^{k+1} \frac{3^{2 k-1}}{k^{2}+1}\) Equation Transcription: Text Transcription: sum_{k=1}^{infty} (-1)^k+1 frac{3^2k -1}{k^2 + 1}

2932 TrueFalse Determine whether the statement is true or false. Explain your answer. An alternating series is one whose terms alternate between even and odd.

2932 TrueFalse Determine whether the statement is true or false. Explain your answer. If a series satisfies the hypothesis of the alternating series test, then the sequence of partial sums of the series oscillates between overestimates and underestimates for the sum of the series.

2932 TrueFalse Determine whether the statement is true or false. Explain your answer. If a series converges, then either it converges absolutely or it converges conditionally.

True–False Determine whether the statement is true or false. Explain your answer. If \(\sum\left(u_{k}\right)^{2}\) converges, then \(\sum u_{k}\) converges absolutely. Equation Transcription: Text Transcription: sum (u_k)^2 sum u_k

Each series satisfies the hypotheses of the alternating series test. For the stated value of \(n\), find an upper bound on the absolute error that results if the sum of the series is approximated by the \(n\)th partial sum. \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k} ; \quad n=7\) Equation Transcription: Text Transcription: n n sum_{k=1}^{infty} frac{(-1)^k+1}{k} ; n=7

Each series satisfies the hypotheses of the alternating series test. For the stated value of \(n\), find an upper bound on the absolute error that results if the sum of the series is approximated by the \(n\)th partial sum. \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k !} ; \quad n=5\) Equation Transcription: Text Transcription: n n sum_{k=1}^{infty} frac{(-1)^k + 1}{k!} ; n=5

Each series satisfies the hypotheses of the alternating series test. For the stated value of \(n\), find an upper bound on the absolute error that results if the sum of the series is approximated by the \(n\)th partial sum. \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{\sqrt{k}} ; \quad n=99\) Equation Transcription: Text Transcription: n n sum_{k=1}^{infty} frac{(-1)^k + 1}{sqrt k} ; n = 99

Each series satisfies the hypotheses of the alternating series test. For the stated value of \(n\), find an upper bound on the absolute error that results if the sum of the series is approximated by the \(n\)th partial sum. \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{(k+1) \ln (k+1)} ; \quad n=3\) Equation Transcription: Text Transcription: n n sum_{k=1}^{infty} frac{(-1)^k+1}{(k+1) ln (k+1)}; n = 3

Each series satisfies the hypotheses of the alternating series test. Find a value of \(n\) for which the \(n\)th sum is ensured to approximate the sum of the series to the stated accuracy. \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k} ; \quad \text { |error } \mid<0.0001\) Equation Transcription: |error| < 0.0001 Text Transcription: n n sum_{k=1}^{infty} frac{(-1)^k+1}{k} ; |error| < 0.0001

Each series satisfies the hypotheses of the alternating series test. Find a value of \(n\) for which the \(n\)th sum is ensured to approximate the sum of the series to the stated accuracy. \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k !} ; \quad \text { |error } \mid<0.0001\) Equation Transcription: |error| < 0.0001 Text Transcription: n n sum_{k=1}^{infty} frac{(-1)^k+1}{k !}; |error| < 0.0001

Each series satisfies the hypotheses of the alternating series test. Find a value of \(n\) for which the \(n\)th sum is ensured to approximate the sum of the series to the stated accuracy. \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{\sqrt{k}}\); two decimal places Equation Transcription: two decimal places Text Transcription: n n sum_{k=1}^{infty} frac{(-1)^k+1}{sqrt k}; two decimal places

Each series satisfies the hypotheses of the alternating series test. Find a value of \(n\) for which the \(n\)th sum is ensured to approximate the sum of the series to the stated accuracy. \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{(k+1) \ln (k+1)} ;\) one decimal place Equation Transcription: one decimal place Text Transcription: n n sum_{k=1}^{infty} frac{(-1)^k+1}{(k+1) ln (k+1)}; one decimal place

Find an upper bound on the absolute error that results if \(s_{10}\) is used to approximate the sum of the given geometric series. Compute \(s_{10}\) rounded to four decimal places and compare this value with the exact sum of the series. \(\frac{3}{4}-\frac{3}{8}+\frac{3}{16}-\frac{3}{32}+\cdots\) Equation Transcription: Text Transcription: s_10 s_10 frac{3}{4} - frac{3}{8} + frac{3}{16} - frac{3}{32} cdot cdot cdot

Find an upper bound on the absolute error that results if \(s_{10}\) is used to approximate the sum of the given geometric series. Compute \(s_{10}\) rounded to four decimal places and compare this value with the exact sum of the series. \(1-\frac{2}{3}+\frac{4}{9}-\frac{8}{27}+\) Equation Transcription: Text Transcription: s_10 s_10 1 - frac{2}{3} + frac{4}{9} - frac{8}{27} + cdot cdot cdot

Each series satisfies the hypotheses of the alternating series test. Approximate the sum of the series to two decimal-place accuracy. \(1-\frac{1}{3 !}+\frac{1}{5 !}-\frac{1}{7 !}+\cdots\) Equation Transcription: Text Transcription: 1 - frac{1}{3!} + frac{1}{5!} - frac{1}{7!} + cdot cdot cdot

Each series satisfies the hypotheses of the alternating series test. Approximate the sum of the series to two decimal-place accuracy. \(1-\frac{1}{2 !}+\frac{1}{4 !}-\frac{1}{6 !}+\cdots\) Equation Transcription: Text Transcription: 1 - frac{1}{2!} + frac{1}{4!} - frac{1}{6!} + cdot cdot cdot

Each series satisfies the hypotheses of the alternating series test. Approximate the sum of the series to two decimal-place accuracy. \(\frac{1}{1.2}-\frac{1}{2.2^{2}}+\frac{1}{3.2^{3}}-\frac{1}{4.2^{4}}+\cdots\) Equation Transcription: Text Transcription: \frac{1}{1.2} - frac{1}{2.2^2} +frac{1}{3.2^3} - frac{1}{4.2^4} + cdot cdot cdot

Each series satisfies the hypotheses of the alternating series test. Approximate the sum of the series to two decimal-place accuracy. \(\frac{1}{1^{5}+4.1}-\frac{1}{3^{5}+4.3}+\frac{1}{5^{5}+4.5}-\frac{1}{7^{5}+4.7}+\cdots\) Equation Transcription: Text Transcription: frac{1}{1^5 + 4.1} - frac{1}{3^5 + 4.3} +frac{1}{5^5 + 4.5} - frac{1}{7^5 +4.7} + cdot cdot cdot

The purpose of this exercise is to show that the error bound in part \((b)\) of Theorem 9.6.2 can be overly conservative in certain cases. (a) Use a CAS to confirm that \(\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots\) (b) Use the CAS to show that \(\left|(\pi / 4)-s_{25}\right|<10^{-2}\). (c) According to the error bound in part \((b)\) of Theorem 9.6.2, what value of \(n\) is required to ensure that \(\left|(\pi / 4)-s_{n}\right|<10^{-2}\)? Equation Transcription: Text Transcription: (b) frac{pi}{4} = 1 - frac{1}{3} + frac{1}{5} - frac{1}{7} + cdot cdot cdot cdot |(pi/4) -s_25| <10^-2 (b) n |(pi/4) - s_n|<10^-2

Prove: If a series \(\sum a_{k}\) converges absolutely, then the series \(\sum a_{k}^{2}\) converges. Equation Transcription: Text Transcription: sum a_k sum a_k^2

(a) Find examples to show that if \(\sum a_{k}\) converges, then \(\sum a_{k}^{2}\) may diverge or converge. (b) Find examples to show that if \(\sum a_{k}^{2}\) converges, then \(\sum a_{k}\) may diverge or converge. Equation Transcription: Text Transcription: sum a_k sum a_k^2 sum a_k^2 sum a_k

Let \(\sum u_{k}\) be a series and define series \(\sum p_{k}\) and \(\sum q_{k}\) so that \(Pk =\left\{\begin{array}{ll}u_{k,} & u_{k}>0 \\0, & u_{k} \leq 0\end{array}\right.\) and \(q k=\left\{\begin{array}{cc}0 & u_{k} \geq 0 \\-u_{k}, & u_{k}<0\end{array}\right.\) (a) Show that \(\sum u_{k}\) converges absolutely if and only if \(\sum p_{k}\) and \(\sum q_{k}\) both converge. (b) Show that if one of \(\sum p_{k}\) or \(\sum q_{k}\) converges and the other diverges, then \(\sum u_{k}\) diverges. (c) Show that if \(\sum u_{k}\) converges conditionally, then both \(\sum p_{k}\) and \(\sum q_{k}\) diverge. Equation Transcription: Text Transcription: sum u_k sum p_k sum q_k Pk= {u_k, & u_{k}>0 0, & u_{k} leq 0 qk = {0 & u_{k} geq 0 -u_{k}, & u_{k} < 0 sum u_k sum p_k sum q_k sum p_k sum q_k sum u_k sum u_k sum p_k sum q_k

It can be proved that the terms of any conditionally convergent series can be rearranged to give either a divergent series or a conditionally convergent series whose sum is any given number \(S\) For example, we stated in Example 2 that \(\ln 2=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\cdots\) Show that we can rearrange this series so that its sum is \(\frac{1}{2} \ln 2\) by rewriting it as \(\left(1-\frac{1}{2}-\frac{1}{4}\right)+\left(\frac{1}{3}-\frac{1}{6}-\frac{1}{8}\right)+\left(\frac{1}{5}-\frac{1}{10}-\frac{1}{12}\right)+\cdots\) [Hint: Add the first two terms in each grouping.] Equation Transcription: Text Transcription: S ln 2 = 1 - frac{1}{2} + frac{1}{3} - frac{1}{4} + frac{1}{5} - frac{1}{6} + cdot cdot cdot frac{1}{2} ln 2 (1- frac{1}{2} - frac{1}{4}) + (frac{1}{3} - frac{1}{6} - frac{1}{8}) + (frac{1}{5} - frac{1}{10} - frac{1}{12}) + cdot cdot cdot

Exercise 51 illustrates that one of the nuances of "conditional" convergence is that the sum of a series that converges conditionally depends on the order that the terms of the series are summed. Absolutely convergent series are more dependable, however. It can be proved that any series that is constructed from an absolutely convergent series by rearranging the terms will also be absolutely convergent and has the same sum as the original series. Use this fact together with parts \((a)\) and \((b)\) of Theorem 9.4.3 in these exercises. It was stated in Exercise 35 of Section 9.4 that \(\frac{\pi^{2}}{6}=1+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}+\cdots\) Use this to show that \(\frac{\pi^{2}}{8}=1+\frac{1}{3^{2}}+\frac{1}{5^{2}}+\frac{1}{7^{2}}+\cdots\) Equation Transcription: Text Transcription: a b frac{pi^2}{6} = 1 + frac{1}{2^2} + frac{1}{3^2} + frac{1}{4^2} + cdot cdot cdot frac{pi^2}{8} = 1 + frac{1}{3^2} + frac{1}{5^2} + frac{1}{7^2} + cdot cdot cdot

Exercise 51 illustrates that one of the nuances of "conditional" convergence is that the sum of a series that converges conditionally depends on the order that the terms of the series are summed. Absolutely convergent series are more dependable, however. It can be proved that any series that is constructed from an absolutely convergent series by rearranging the terms will also be absolutely convergent and has the same sum as the original series. Use this fact together with parts \((a)\) and \((b)\) of Theorem 9.4.3 in these exercises. Use the series for \(\pi^{2} / 6\) given in the preceding exercise to show that \(\frac{\pi^{2}}{12}=1-\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}+\cdots\) Equation Transcription: Text Transcription: a b pi^2/6 frac{pi^2}{12} = 1 + frac{1}{2^2} + frac{1}{3^2} + frac{1}{4^2} + cdot cdot cdot

Exercise 51 illustrates that one of the nuances of "conditional" convergence is that the sum of a series that converges conditionally depends on the order that the terms of the series are summed. Absolutely convergent series are more dependable, however. It can be proved that any series that is constructed from an absolutely convergent series by rearranging the terms will also be absolutely convergent and has the same sum as the original series. Use this fact together with parts \((a)\) and \((b)\) of Theorem 9.4.3 in these exercises. It was stated in Exercise 35 of Section 9.4 that \(\frac{\pi^{4}}{90}=1+\frac{1}{2^{4}}+\frac{1}{3^{4}}+\frac{1}{4^{4}}+\cdots\) Use this to show that \(\frac{\pi^{4}}{96}=1+\frac{1}{3^{4}}+\frac{1}{5^{4}}+\frac{1}{7^{4}}+\cdots\) Equation Transcription: Text Transcription: a b frac{pi^4}{90} = 1 + frac{1}{2^4} + frac{1}{3^4} + frac{1}{4^4} + cdot cdot cdot frac{pi^4}{96} = 1 + frac{1}{3^4} + frac{1}{5^4} + frac{1}{7^4} + cdot cdot cdot

Writing Consider the series \(1-\frac{1}{2}+\frac{2}{3}-\frac{1}{3}+\frac{2}{4}-\frac{1}{4}+\frac{2}{5}-\frac{1}{5}+\cdots\) Determine whether this series converges and use this series as an example in a discussion of the importance of hypotheses \((a)\) and \((b)\) of the alternating series test (Theorem 9.6.1). Equation Transcription: Text Transcription: 1 - frac{1}{2} + frac{2}{3} - frac{1}{3} + frac{2}{4} - frac{1}{4} + frac{2}{5} - frac{1}{5} + cdot cdot cdot a b

Writing Discuss the ways that conditional convergence is "conditional." In particular, describe how one could rearrange the terms of a conditionally convergent series \(\sum u_{k}\) so that the resulting series diverges, either to \(+\ \infty\) or to \(-\ \infty\). [Hint: See Exercise 50.] Equation Transcription: Text Transcription: sum u_{k} + infty - infty